Category Theory and Topos Theory, Spring 2018

This course is part of the Mastermath programme.
Lecturer is Jaap van Oosten. Assistant for the exercise class is Mark Kamsma (mark@markkamsma.nl).

This course will be given in weeks 6-20 (February-June 2018).
The course is on Mondays, 10:15-12:00 (lecture) and 12:15-13:00 (exercises), in Amsterdam Science Park, room A1.04.
The course will be concluded by a written exam, for which you must obtain at least the grade 5. There are, during the course, 6 hand-in exercises, which may count for 30% of your final grade (it is not compulsory, although strongly recommended, to hand in these exercises). That is, your final grade will be calculated as follows: let H be your grade for the hand-in exercises and W your grade for the written exam. Provided W is at least 5, your final grade will be the maximum of W and the number (3H + 7W)/10.
The hand-in exercises have to be handed in at the exercise class in the week they are mentioned below; they can also be sent by email to the assistant, before noon on the same day. Exercises, to be handed in in week x+1, will appear on this page ultimately the Wednesday of week x.
Here are the hand-in exercises with solutions for the old ones.

Literature: Lecture Notes.
This course will treat some applications of Category Theory to Logic. Although this is only a minor part of the course, people unfamiliar with Logic may wish to consult sections 2.1-2.3 of the basic logic course Sets, Models and Proofs, for the notions of a language and structures for a language.
Suggestions for improvement of the lecture notes are very welcome!

Making diagrams in LaTeX

Mark Kamsma prefers this tool. He gives the following instructions:
Draw your diagram using that tool, and then use the button with the braces { } to copy the LaTeX code for your diagram. You can then paste the code in a tikzcd environment, or more concrete:
\[
\begin{tikzcd}
Paste your code here
\end{tikzcd}
\]
Do not forget to include \usepackage{tikz-cd} in your LaTeX document to use the tikzcd environment.

Overview of the material treated in the course

• Week 6: Chapter 1: Categories and Functors, examples, special objects and arrows.
  Exercises: 1,4,5,7,9.
• Week 7: Chapter 2: Natural Transformations
  exercises: 20,22,27,32,33.
• Week 8: Chapter 3: Limits and Colimits
  Exercises:34,35,36,38,44.
  Hand-in exercise: 1
• Week 9: Chapter 4: Regular Categories and a first glimpse at Categorical Logic.
  Exercises: 65,66,67,70.
• Week 10: More on Regular Categories and Regular Logic.
  Exercises: 75,79,88,89,92.
  Hand-in exercise: 2
• Week 11: Adjunctions.
  Exercises: 95,98,99,103.
• Week 12: Monads and Algebras.
  Exercises: 111,113,117,118,123.
  Hand-in exercise: 3
• Week 13: Cartesian Closed Categories and Natural Numbers Objects.
  Exercises: 125,126,130,132.
• Week 14: Easter Monday. No lecture.
• Week 15: Presheaf categories, cartesian closed structure, every object a colimit of representable presheaves, subobject classifier.
  Exercises: 144,145,146,149 and: Calculate the subobject classifier in the category of presheaves over a monoid M.
  Hand-in exercise: 4
• Week 16: Continuation of theory of presheaves; sections 9.1 and 9.2.
  Exercises: 152,153,154,157.
• Week 17: Logic of presheaves; definition of Grothendieck topology, separated presheaves and sheaves.
  Exercises: 158,160,161,162.
  Hand-in exercise: 5
• Week 18: rest of 10.1, 10.2
  Exercises: 164,165,167,168.
• Week 19: 10.3 and 10.4.
  Exercises: 176,177.
  Hand-in exercise: 6. This exercise may be sent in until May 9, midnight.
• Week 20: Further perspectives: morphisms of toposes.
  Left-over exercises; questions.
• Week 25: Exam, Monday June 18, 10:00--13:00, SP C0.05.
  At the exam, the use of lecture notes, books, or any kind of device which can store information, is not allowed.
  Here is the exam, with solutions.
• Week 27: Retake Exam, Monday July 2, 10:00--13:00, SP B0.204.
  Here is the exam of 2014, with solutions.
  Here is the retake exam of 2014, without solutions.
  Here is the exam of 2016, with solutions

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